The Existence Of Two Strictly Disjoint Pure Idempotent Latin Squares

Let A be a Latin square on a v-set S which is a v×v array such that each of the v symbols occurs once in each row and each column,respectively.A is said to be a pure idempotent Latin square,if there are v(v-1)3-tuples in the set {{i,j,a_ij}:i,j∈S and i,j,a_ijare different} and there exists a_ii=i for each i∈S.Two pure idempotent Latin squares are strictly disjoint,if their 3-tuple sets are disjoint.Great progress has been made on the various large sets problems and overlarge sets problems since the LKTS(15)was presented by Sylvester in 1850.The large sets of pure idempotent Latin square is a part of large sets of Latin squares.So far,the existence spectrum of pure idempotent Latin squares has been established.In this article,we focus on the existence of two strictly disjoint pure idempotent Latin squares.This thesis is organized as follows.In Chapter 1,some basic conceptions are introduced,and some basic theorems are given,which are useful in Chapter 2 and Chapter 3.In Chapter 2,two strictly disjoint pure idempotent Latin squares with prime power orders are obtained on Galois field.And we also prove that there exist[(p-2)/6]×6 pairwise disjoint pure idempotent Latin squares,where p is a prime power order and p≥8.In Chapter 3,we discuss the problem of two strictly disjoint pure idempotent Latin squares with non-prime power orders.Applying the relationship between pure idempotent Latin squares and incomplete pure idempotent Latin squares,the problem will turn to that of searching for two strictly disjoint incomplete pure idempotent Latin squares.Direct constructions and recursive constructions for incomplete pure idempotent Latin square are presented.Using these constructions,we get the following main result:if v≡0,1,2,3,4,8,10,12,14(mod 15)and v≥14,v {14,30,42,44,45,46,48,55,57,60,62,63,92,98},then there exist two strictly disjoint pure idempotent Latin squares.In Chapter 4,more problems on pure idempotent Latin squares are given which need to be solved.Finally,the existence order of two strictly disjoint pure idempotent Latin squares below 100 are listed.